Start your trial now! First week only $4.99!*arrow_forward*

BuyFind*launch*

4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.7, Problem 52E

To determine

**To find:** The points where the greatest integer function

differentiable and then to find the formula for

Expert Solution

The greatest integer function is not differentiable at

The formula for the derivative function is

**Result Used:**

The greatest integer function is defined as follows,

**Calculation:**

Obtain the points where the greatest integer function

The derivative of the greatest integer function is calculated as follows,

**Case 1:** *x* is not an integer.

Since *x* is not an integer,

Thus,

**Case 2:** *x* is an integer.

The left hand derivative of the function is computed as follows,

Since *h* approaches 0 from left,

This implies that,

Thus, the left hand derivative of the function does not exist. This follows that,

Therefore, the greatest integer function is not differentiable at the integer points.

By case (1) and (2), it is defined that the formula for the derivative is

**Graph:**

Use the above information and trace the graph of

From the Figure 1, it is clear that the greatest integer function is discontinuous at integer points.